We start therefore by recalling the characterization of the cofibrant objects in Lack's model structure, which requires the notion of free categories. Recall that a graph is a specified by a pair of functions with the same domain and the same codomain. Every small category has an underlying graph, given by the source and target functions on the set of morphisms, i.e., there is a forgetful functor \(U:{C}\mathrm {at}\rightarrow {G}\mathrm {rph}\) . The forgetful functor admits a left adjoint \({F}_1:{G}\mathrm {rph}\rightarrow {C}\mathrm {at}\) , which takes a graph to its free category. For more details regarding free categories and graphs see [1]}. The following result regarding cofibrant objects summarizes [2]}.
